Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and s, it follows constructively that the K-completeness of s implies MP(S), a form of Markov's Principle. If s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when s is many-one complete, MP(S) implies the usual Markov's Principle MP.An immediate corollary is that the Tarski, Beth and Kripke weak completeness theorems for the negative fragment of intuitionistic predicate logic are unobtainable in HAS. Second, each of these: weak completeness for classical predicate logic, weak completeness for the negative fragment of intuitionistic predicate logic and strong completeness for sentential logic implics MP. Beth and Kripke completeness for intuitionistic predicate or sentential logic also entail MP.These results give extensions of the theorem of Gödel and Kreisel (in [4]) that completeness for pure intuitionistic predicate logic requires MP. The assumptions of Gödel and Kreisel's original proof included the Axiom of Dependent Choice and Herbrand's Theorem, no use of which is explicit in the present article.     

Relevant Robinson's arithmetic

In this paper two different formulations of Robinson's arithmetic based on relevant logic are examined. The formulation based on the natural numbers (including zero) is shown to collapse into classical Robinson's arithmetic, whereas the one based on the positive integers (excluding zero) is shown not to similarly collapse. Relations of these two formulations to R. K. Meyer's system R# of relevant Peano arithmetic are examined, and some remarks are made about the role of constant functions (e.g., multiplication by zero) in relevant arithmetic.This paper has been greatly influenced by the (largely unpublished) work of E. K. Meyer (cf. [7]) on relevant arithmetic, and I wish to thank him, and also N. D. Belnap, Jr. and D. Cohen for helpful advice. In fairness to Meyer it must be said that he finds my axioms 13 and 13(1) too strong (they are not theorems of his system R# - cf. §4 below). Meyer tells me be finds vindication for his view in my chief theorem of §2. For myself, I find the insights behind Meyer's work on R# to be both stable and fruitful, and if I now had to make a choice, I would follow Meyer in his rejection of my axioms. However, the systems I explore in this paper themselves have a surprising amount of internal consistency of motivation (cf. §5). Let a hundred formal systems bloom.     

A Cut-Free Sequent System for the Smallest Interpretability Logic

The idea of interpretability logics arose in Visser [Vis90]. He introduced the logics as extensions of the provability logic GLwith a binary modality . The arithmetic realization of A B in a theory T will be that T plus the realization of B is interpretable in T plus the realization of A (T + A interprets T + B). More precisely, there exists a function f (the relative interpretation) on the formulas of the language of T such that T + B C implies T + A f(C).The interpretability logics were considered in several papers. An arithmetic completeness of the interpretability logic ILM, obtained by adding Montagna's axiom to the smallest interpretability logic IL, was proved in Berarducci [Ber90] and Shavrukov [Sha88] (see also Hájek and Montagna [HM90] and Hájek and Montagna [HM92]). [Vis90] proved that the interpretability logic ILP, an extension of IL, is also complete for another arithmetic interpretation. The completeness with respect to Kripke semantics due to Veltman was, for IL, ILMand ILP, proved in de Jongh and Veltman [JV90]. The fixed point theorem of GLcan be extended to ILand hence ILMand ILP(cf. de Jongh and Visser [JV91]). The unary pendant "T interprets T + A" is much less expressive and was studied in de Rijke [Rij92]. For an overview of interpretability logic, see Visser [Vis97], and Japaridze and de Jongh [JJ98].In this paper, we give a cut-free sequent system for IL. To begin with, we give a cut-free system for the sublogic IL4of IL, whose -free fragment is the modal logic K4. A cut-elimination theorem for ILis proved using the system for IK4and a property of Löb's axiom.     

A Negationless Interpretation of Intuitionistic Theories. I

The present work contains an axiomatic treatment of some parts of the restricted version of intuitionistic mathematics advocated by G. F. C. Griss, also known as negationless intuitionistic mathematics. Formal systems NPC, NA, and FIM ^N for negationless predicate logic, arithmetic, and analysis are proposed. Our Theorem 4 in Section 2 asserts the translatability of Heyting's arithmetic HAinto NA. The result can in fact be extended to a large class of intuitionistic theories based on HAand their negationless counterparts. For instance, in Section 3 this is shown for Kleene's system of intuitionistic analysis FIMand our FIM ^N . This revised version was published online in June 2006 with corrections to the Cover Date.     

Arithmetic,Mathematical Intuition,and Evidence

Abstract

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic (Oxford: Oxford University Press, 2011). The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics.     

Heuristics and biases in mental arithmetic: revisiting and reversing operational momentum

ABSTRACT

Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3?0) but reverse OM with non-zero problems (e.g., 2+1, 4?1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving.     

Mathematics anxiety and mental arithmetic performance: An exploratory investigation

Abstract

Two exploratory studies were conducted to determine if mathematics anxiety, as assessed by the Mathematics Anxiety Rating Scale (MARS), is related to the underlying mental processes of arithmetic performance. MARS scores were higher when the test was administered by computer, vs. the standard paper-and-pencil format, and were higher for female than male college students. Small but significant processing differences in simple addition and multiplication were found when subjects were divided by quartiles into anxiety groups. Much larger differences in processing speed and accuracy were found with complex addition problems and a set of difficult problems (e.g. 9 × 16 = 134, true or false) that tested all four arithmetic operations. Overall, the low anxiety group was consistently the most rapid and accurate, the medium high was consistently the slowest, and the high anxiety group the most prone to errors. The results suggest that genuine performance differences exist among the several levels of mathematics anxiety, and that chronometric, reaction time-based studies of such performance will be useful in revealing those differences.     

Nelson algebras through Heyting ones: I

The main aim of the present paper is to explain a nature of relationships exist between Nelson and Heyting algebras. In the realization, a topological duality theory of Heyting and Nelson algebras based on the topological duality theory of Priestley ([15], [16]) for bounded distributive lattices are applied. The general method of construction of spaces dual to Nelson algebras from a given dual space to Heyting algebra is described (Thm 2.3). The algebraic counterpart of this construction being a generalization of the Fidel-Vakarelov construction ([6], [25]) is also given (Thm 3.6). These results are applied to compare the equational category N of Nelson algebras and some its subcategories (and their duals) with the equational category H of Heyting algebras (and its dual). It is proved (Thm 4.1) that the category N is topological over the category H. The main results of this article are a part of theses of the author's doctoral dissertation at the Nicholas Copernicus University in 1984 (cpmp. [24]).Research partially supported by Polish Government Grant CPBP 08-15.     

Axiomatization and completeness of uncountably valued approximation logic

A first order uncountably valued logicL _Q(0,1) for management of uncertainty is considered. It is obtained from approximation logicsL _T of any poset type (T, ) (see Rasiowa [17], [18], [19]) by assuming (T, )=(Q(0, 1), ) — whereQ(0, 1) is the set of all rational numbersq such that 0<q<1 and is the arithmetic ordering — by eliminating modal connectives and adopting a semantics based onLT-fuzzy sets (see Rasiowa and Cat Ho [20], [21]). LogicL _Q(0,1) can be treated as an important case ofLT-fuzzy logics (introduced in Rasiowa and Cat Ho [21]) for (T, )=(Q(0, 1), ), i.e. asLQ(0, 1)-fuzzy logic announced in [21] but first examined in this paper.L _Q(0,1) deals with vague concepts represented by predicate formulas and applies approximate truth-values being certain subsets ofQ(0, 1). The set of all approximate truth-values consists of the empty set ø and all non-empty subsetss ofQ(0, 1) such that ifqs andqq, thenqs. The setLQ(0, 1) of all approximate truth-values is uncountable and covers up to monomorphism the closed interval [0, 1] of the real line.LQ(0, 1) is a complete set lattice and therefore a pseudo-Boolean (Heyting) algebra. Equipped with some additional operations it is a basic plain semi-Post algebra of typeQ(0, 1) (see Rasiowa and Cat Ho [20]) and is taken as a truth-table forL _Q(0,1) logic.L _Q(0,1) can be considered as a modification of Zadeh's fuzzy logic (see Bellman and Zadeh [2] and Zadeh and Kacprzyk, eds. [29]). The aim of this paper is an axiomatization of logicL _Q(0,1) and proofs of the completeness theorem and of the theorem on the existence ofLQ(0, 1)-models (i.e. models under the semantics introduced) for consistent theories based on any denumerable set of specific axioms. Proofs apply the theory of plain semi-Post algebras investigated in Cat Ho and Rasiowa [4].Presented byCecylia Rauszer     

Swedish youth football players’ attitudes towards moral decision in sport as predicted by the parent-initiated motivational climate

ObjectivesThe purpose of this study was to examine associations between late adolescent football players’ perceptions of the motivational climate – as initiated by mothers and fathers – and attitudes towards moral decision making in sports.DesignCross-sectional.MethodsParticipants were 213 Swedish football players (144 males, 67 females) aged 16–19 years who completed measures assessing perceived parent-initiated motivational climate (i.e., success-without-effort climate [SWEC]; worry conducive climate [WCC]; and learning/enjoyment climate [LEC]) and attitudes towards moral decision-making in sport (i.e., acceptance of cheating [AOC]; acceptance of gamesmanship [AOG] and keeping winning in proportion [KWIP]).ResultsCanonical correlations demonstrated moderate positive relations between parent-initiated − both mother and father − performance climates (WCC and SWEC) and AOC and AOG. Moreover, the relationship between mother and father-initiated learning/enjoyment climate (LEC) were shown to be moderately and positively associated with the prosocial attitude dimension of KWIP. Results also showed that a mother-initiated LEC and a mother-initiated SWEC were stronger predictors of the criterion variables (AOC, AOG, and KWIP) than equivalent father-initiated climate dimensions.ConclusionsThe results highlight the importance of considering the relationship between parent-initiated climates − especially initiated by mothers − and the development of moral decision-making among youth football players.     

Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities

Abstract

Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N?=?709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.     

Reversed Resolution in Reducing General Satisfiability Problem

In the following we show that general property S considered by Cowen [1], Cowen and Kolany in [3] and earlier by Cowen in [2] and Kolany in [4] as hypergraph satisfiability, can be constructively reduced to (3, 2) · SAT, that is to satisfiability of (at most) triples with two-element forbidden sets. This is an analogue of the“classical” result on the reduction of SAT to 3 · SAT.     

On the Existence of Continua of Logics Between Some Intermediate Predicate Logics

A method for constructing continua of logics squeezed between some intermediate predicate logics, developed by Suzuki [8], is modified and applied to intervals of the form [L, L+ ¬¬S], where Lis a predicate logic, Sis a closed predicate formula. This solves one of the problems from Suzuki's paper.     

On the definability of the quantifier “there exist uncountably many”

In paper [5] it was shown that a great part of model theory of logic with the generalized quantifier Q x = there exist uncountably many x is reducible to the model theory of first order logic with an extra binary relation symbol. In this paper we consider when the quantifier Q x can be syntactically defined in a first order theory T. That problem was raised by Kosta Doen when he asked if the quantifier Q x can be eliminated in Peano arithmetic. We answer that question fully in this paper.I would like to thank Kosta Doen and Zoran Markovi who made valuable suggestions and remarks on a draft of this paper.     

Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective

ABSTRACT

Solutions to word problems are moderated by the semantic alignment of real-world relations with mathematical operations. Categorical relations between entities (tulips, roses) are aligned with addition, whereas certain functional relations between entities (tulips, vases) are aligned with division. Similarly, discreteness vs. continuity of quantities (marbles, water) is aligned with different formats for rational numbers (fractions and decimals, respectively). These alignments have been found both in textbooks and in the performance of college students in the USA and in South Korea. The current study examined evidence for alignments in Russia. Textbook analyses revealed semantic alignments for arithmetic word problems, but not for rational numbers. Nonetheless, Russian college students showed semantic alignments both for arithmetic operations and for rational numbers. Since Russian students exhibit semantic alignments for rational numbers in the absence of exposure to examples in school, such alignments likely reflect intuitive understanding of mathematical representations of real-world situations.     

Cardiovascular reactions to psychological stress and abuse history: the role of occurrence,frequency, and type of abuse

Background and objectives: Extreme cardiovascular reactions to psychological stress have been associated with traumatic life experiences. Previous studies have focused on the occurrence or frequency of abuse rather than type of abuse. We examined how occurrence, frequency, and the type of abuse history are related to cardiovascular reactivity (CVR) to acute psychological stress. Design: The study consisted of between group and continuous analyses to examine the association between occurrence, type, and frequency of abuse with cardiovascular reactions to acute psychological stress. Methods: Data from 64 participants were collected. Heart rate, systolic blood pressure, and diastolic blood pressure were measured at baseline and during a standard mental arithmetic stress task. Results: Individuals who experienced abuse showed diminished CVR to acute psychological stress; this was driven specifically by the history of sexual abuse. Frequency of abuse did not relate to stress reactions. Conclusions: These findings accord with previous work suggesting a relationship between traumatic life experience and hypoarousal in physiological reactivity and extend previous findings by suggesting the relationship may be driven by sexual abuse.     

Saving-enhanced performance: saving items after study boosts performance in subsequent cognitively demanding tasks

ABSTRACT

By saving and storing information, we use digital devices as our external memory stores, being able to offload and temporarily forget saved contents. Storm and Stone [2015. Saving-enhanced memory: The benefits of saving on the learning and remembering of new information. Psychological Science, 26(2), 182–188] showed that such memory offloading can be beneficial for subsequent memory performance. Saving already encoded items enhanced recall of items encoded after saving. In the present study, we did not only replicate saving-enhanced memory but found saving-enhanced performance for unrelated cognitively demanding tasks. Participants solved more modular arithmetic problems when they were able to offload a previously studied word list, compared to trials without the possibility to offload. Thus, saving of recently encoded items entailed a general benefit on subsequent cognitive performance, beyond encoding and retrieving word lists. We assume that offloading frees participants from the need to maintain offloaded items. Gained resources can then be used for subsequent tasks with high cognitive demands. In a nutshell, memory offloading can help to reduce the amount of information that has to be processed at a given time, efficiently delegating our limited cognitive resources to the most relevant tasks at hand while currently irrelevant information are safely stored outside our own memory.     

Glivenko Type Theorems for Intuitionistic Modal Logics

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising.     

Performance feedback,self-esteem,and cardiovascular adaptation to recurring stressors

Background and Objectives: This study sought to examine the effects of performance feedback and individual differences in self-esteem on cardiovascular habituation to repeat stress exposure.

Methods: Sixty-six university students (n?=?39 female) completed a self-esteem measure and completed a cardiovascular stress-testing protocol involving repeated exposure to a mental arithmetic task. Cardiovascular functioning was sampled across four phases: resting baseline, initial stress exposure, a recovery period, and repeated stress exposure. Participants were randomly assigned to receive fictional positive feedback, negative feedback, or no feedback following the recovery period.

Results: Negative feedback was associated with a sensitized blood pressure response to a second exposure of the stress task. Positive feedback was associated with decreased cardiovascular and psychological responses to a second exposure. Self-esteem was also found to predict reactivity and this interacted with the type of feedback received.

Conclusions: These findings suggest that negative performance feedback sensitizes cardiovascular reactivity to stress, whereas positive performance feedback increases both cardiovascular and psychological habituation to repeat exposure to stressors. Furthermore, an individual’s self-esteem also appears to influence this process.     

Some paraconsistent sentential calculi

In [8] Jakowski defined by means of an appropriate interpretation a paraconsistent calculusD ₂ . In [9] J. Kotas showed thatD ₂ is equivalent to the calculusM(S5) whose theses are exactly all formulasa such thatMa is a thesis ofS5. The papers [11], [7], [3], and [4] showed that interesting paraconsistent calculi could be obtained using modal systems other thanS5 and modalities other thanM. This paper generalises the above work. LetA be an arbitrary modality (i.e. string ofM's,L's and negation signs). Then theA-extension of a set of formulasX is {¦A X}}. Various properties ofA-extensions of normal modal systems are examined, including a problem of their axiomatizability